Superconducting radio-frequency (SRF) cavities are useful in a variety of modern applications, such as free-electron lasers and particle colliders [1]}. The primary advantage of SRF cavities over their normal-conducting alternatives is a lower surface resistance, which has the obvious benefit of a lower energy footprint [2]}. Despite decades of steady improvements, allowing for quality factors in the neighborhood of \(10^{10}\)  [1]} and accelerating fields up to \(25-45\)  MV/m [2]}, some unanswered questions remain. At high fields there is often a “\(Q\)  slope,” meaning the quality factor decreases—or, equivalently, surface resistance increases—as field strength increases. More confounding, recently some cavities, in particular niobium cavities doped with nitrogen, have exhibited an “anti-\(Q\)  slope,” meaning that the quality factor actually increases with field [1]}. Increasing the resonant frequency of the cavity has been associated with stronger anti-\(Q\)  slopes [6]}. Since inelastic scattering rates [7]} can be comparable to typical SRF frequencies [8]}, this suggests inelastic scattering may play an important role in determining the field dependence of surface resistance.
